OK I swear I'm only coming on for a quick post and then to bed 
Hi drspouse. Firstly I would say never ever feel guilty or worry about shouting as a response to pain or danger - it's a natural response and there for a reason! Definitely not damaging. Shouting is unhelpful if you're doing it in a berating way or as a response to shouting, but you shouldn't censor yourself shouting in pain. IMO.
What was the problem with the counting to 10? My feeling would be if it helps him calm down does it matter if he thinks of it as a reward? Or do you mean he was doing stuff in order to be allowed to count to 10? Could you let him count to 10 any time without the other behaviour to give him the message he doesn't have to do something to be made to count to 10? I don't think that small children actually like doing the hitting/biting/screaming stuff, it's not actually very nice for them either. Imagine how you feel when you get the urge to scream or hit - it's a stressful, unpleasant feeling. And it's not like they do it randomly when they're having fun.
At three it's all about simple. I'd say this: "asking him to calm down, talking calmly about why he's sitting down/repeating reason for holding him/asking him to say what he did and say sorry." is a bit much bombarding him unless the quiet talking is helping him to calm down - lots of info bombarding is probably going to be annoying/and or a bit "information overload" and feed into it. If there is one thing I learned about three year olds it's that when they are in that space you just CAN'T reason or anything with them, and it stresses everyone out to try. Simple, predictable, firm, but you don't need to be harsh.
If you think about it from his position, he's acting out because he doesn't understand that he can't have something or do something or something is wrong. The reason might seem insignificant to us but it's really, really important to them. You can try reflecting his feelings back to him "Oh DS you look very angry and cross. Are you cross about X?" I think that perhaps making him repeat what he has done wrong is counterproductive? It's focusing on the bad behaviour which sounds like it is a loss of control, where he might not really have been fully aware and in control of what he was doing. It's quite a lot to expect a 3 year old to then take responsibility for that (which is what you're doing when asking him to repeat it and say sorry). At this age I used to tell DS the effect (Daddy is hurt now,) and give the suggestion (It might make him feel better if you say sorry and give him a cuddle). It was hard though because actually it was mostly on me and it's hard to discipline and be the victim all at once! One thing which we've found helps when he gets out of control recently, as has happened a couple of times, is to hold him so he's facing away in a sort of hug - he can't hurt you or himself, then. He does and always did awful screaming saying "You're hurting me!" but as long as you're not squeezing really tight, you're not hurting them, and then you can repeat quite calmly in a low pitched voice "I won't let you hurt me or yourself. I know you feel angry. I can't let you hurt me." He is six now though so it really DOES hurt when he hits now! Thankfully extremely rare these days, but we always get the escalation if we try to give consequences too quickly/without warning or react with any kind of force. DH is a bit of one for retaliating quickly as he thinks he should be "in charge" but it doesn't always help.
Also redirection helped a lot with DS when he was little. Saying "You can hit the sofa but not people". And immediate removal of any toy used to hit or about to be used to hit, no questions, no exceptions. Removed until he was really properly calm and had been for a while. (I sometimes used to forget and find random cupboards full of trains that had been missing for months 
)
He sounds very very similar to my DS and I think he does sound sensitive if he's thinking about other DC like that - sensitive doesn't necessarily mean "difficult and odd" - they can be very very pleasant children, especially around settings which aren't home. I'm starting to find a lot of sensitivities with DS that I took for granted or didn't notice before, and I really struggled with the ages of 3-5, but I think I would have struggled less if I was able to actually realise what was going on earlier. I just thought his reactions were normal - and to an extent, they were - I wasn't very good at reacting in a consistent way. I tried too hard to be "gentle" and ended up just being confusing.
Blue I read something the other day which made me laugh - in the middle of an article a woman said "it would be incredible if children could come out already six years old. You could just help them with their issues instead of inventing new ones." So true. Exactly how I feel. Except I'm starting to realise that maybe I'm putting too much pressure on myself and DS would have DS' issues whether I was his mum or not and maybe it is okay to just try and figure them out and our way around them and not fix everything. I mean everyone has issues, right? So many of us worrying about whether there's a diagnosable "thing" when maybe it's just them being their unique selves and it's just that in the past we saw things more simply and a child was just being naughty/lazy/dramatic etc.
MoreSnow 
about the friend, how insensitive. I hate it when people are smug about things. Here's a clue: There is no magical perfect method that works for everyone! Let's hope she has a contraception failure who doesn't sleep through until three years old 
Dreaming I feel like I do too much shaming too. My default actually is criticism. It can help to try and catch yourself doing it and then consciously try and comment on one positive thing even though it feels fake.
Sorry to hear about your stressful holiday. It definitely sounds like he had totally unrealistic expectations. Horrid that a break should end up being so stressful :( It totally pushes the brink of your relationship, this small children parenting stuff. Make or break. I've done both (break with XP, been pushed right to the brink with DH but make with him, obv) after stressful times and it does always work out to be what's right in the end. If you come through it then you're stronger together and if you don't then it wasn't right anyway.
Claire Show me ONE parent who has a perfect relationship with their 13yo?! :)
LetsGo The book is called "The Life Changing Magic of Tidying". General consensus on the thread (in Good Housekeeping) is that Marie Kondo is vv young and doesn't have DC and probably lives alone in a swish tiny Japanese house and so Doesn't Get It, but her advice is oddly calming anyway.
I am now at the point where scrolling up to find a post to respond to is too far and makes my finger hurt so I'm going to sign off now (it's also taken me 40 minutes to write this post 
sorry and hello to anyone I've missed! :)